Report

Away from High Loss in Plasmonics and Towards Low in Loss in Plasmonics and Metamaterials Battling Loss Plasmonics and Metamaterials Metamaterials Jacob B Khurgin Johns Hopkins University RBNI-14 1 5 stages of dealing with the loss 1. Denial 2. Anger 3. Bargaining 4. Despair 5. Acceptance RBNI-14 2 9000 80 8000 70 7000 60 6000 50 ? 5000 40 4000 30 3000 20 2000 10 10,000 BC 1000 (discovery of Ag) 0 1990 Electron scattering time in Ag (fs) Publications in Plasmonics and Metamaterials (Web of Science ) Motivation 1995 2000 RBNI-14 year 2005 2010 2015 3 Scope •Why are the metals necessary for sub-wavelength confinement? •What are the surface plasmons and polaritons? •Why subwavelength confinement in optical range always means high loss? •Why reducing loss is so important? 3 Case studies 1. Who needs negative index? 2. How does loss impact plasmonic enhancement of the emission? 3. Plasmons and nonlinear optics –a winning combination? •Why and how do the metals absorb and reflect? •Can a metal be made lossless? •Does it have to be a metal? •Can metal loss be compensated by gain? •Can one make a true sub-wavelength laser /spaser? RBNI-14 4 Scope •Why are the metals necessary for subwavelength confinement? •What are the surface plasmons and polaritons? •Why subwavelength confinement in optical range always means high loss? •Why reducing loss is so important? 3 Case studies 1. Who needs negative index? 2. How does loss impact plasmonic enhancement of the emission? 3. Plasmons and nonlinear optics –a winning combination? •Why and how do the metals absorb and reflect? •Can a metal be made lossless? •Does it have to be a metal? •Can metal loss be compensated by gain? •Can one make a true sub-wavelength laser /spaser? RBNI-14 5 Why metal? Metal’s claim to fame is its negative real part of the dielectric constant –Drude Formula Plasma frequency p2 ( ) 1 Plasma frequency is defined as 2 j Scattering rate Ne 2 0m 2 p Metal Plasma Frequencies and Scattering Rates Plasma frequency gold silver aluminum Au Ag Al 2081 THz 8.5 eV 2182 THz 9.0 eV 3231 THz 13.2 eV RBNI-14 Scattering rate 12.31013 s-1 3.2 1013 s-1 27.31013 s-1 6 Permittivity of Gold Drude fit experimental Interband absorption K. Busch et al., Phys. Rep. 444, 101 (2007) RBNI-14 7 Material Q-factor Re( ) QM Im( ) Realistically…QM~10-20 because of surface scattering RBNI-14 8 What does negative mean? The electrons move to screen the electro-magnetic field – hence the field gets expelled from the metal and the field inside becomes evanescent E ES - Total Field=0 - + + + + + + + + + + + + Metal Refractive index –complex, mostly imaginary External field Screening field n() () n() j () Evanescent Field E ~ E0e ~ E0e ( ) z c E Surface Charge x j n ( ) z c z Free Electrons RBNI-14 9 Bulk Charge Oscillations ( ) 1 p2 2 j In bulk metal when the frequency exceeds plasma frequency the electrons can no longer follow the electric field and screen it – metal becomes transparent (p ) 0 At plasma frequency the bulk metal supports longitudinal charge oscillations EE + + + + + + + + + + + + ++++++++++++- RBNI-14 10 Why do we need free carriers? We want to concentrate optical field on subwavelength scale…but are prevented by the diffraction limit Lmin ~ / 2n Comes from the uncertainty principle px / 2 k x / 2 2 n Lmin 1/ 2 Lmin ~ / 4 n Easy to understand for real n…but what about imaginary part? To see how we can beat diffraction limit …let us re-derive it from the energy RBNI-14 conservation considerations 11 a Energy balance in a modea t=0 t/2 E H n k E sin(kz)sin(t ) 0 c H cos(kz) cos(t ) E H E nE H = k 0 2 Electric t 2 2 E Magnetic H E “Potential” U e U M 0 “Kinetic” Energy 2 2 2 Energy Energy oscillates between potential (electric) and kinetic (magnetic) 2 n UM UE RBNI-14 12 Lack of energy balance in a sub- mode a 0 2 c a 2n n a t=0 t=0 H E H cos z cos(t ) E sin z sin(t ) a a E a 2a 2na nE H = H E c E t 0 0 0 2 2 Electric E 2 Magnetic 2 2 “Kinetic” U H 2na E 2na U “Potential” U e E M 0 2 2 2 Energy Energy If a<<0/2n there is almost no magnetic field (quasistatic limit) UM<<UE –energy is not conserved UM UE RBNI-14 0 0 The energy will radiate because it cannot all fit into magnetic energy –this is diffraction limit! 13 Free carriers restore balance in a sub- mode a t=0 0 2 c a 2n n t=0 + + + E 2na nE E sin z sin(t ) H a 0 0 Electric “Potential” U e Energy a 2 E 2 v J 2 True Magnetic 2na Kinetic” “Kinetic” U M UE Energy of Energy 0 electrons At some resonant frequency 0 the balance is achieved UE UM RBNI-14 UK H H cos z cos(t ) a J cos(t ) Nmv 2 LK J 2 UK 2 2 ~ 2 LK CU E LK=Kinetic Inductance (inertia of electrons) If a<<0/2n there is almost no magnetic field (quasistatic limit) UM<<UE – hence energy oscillates between electric energy and kinetic 14 energy of free carriers Scope •Why are the metals necessary for sub-wavelength confinement? •What are the surface plasmons and polaritons? •Why subwavelength confinement in optical range always means high loss? •Why reducing loss is so important? 3 Case studies 1. Who needs negative index? 2. How does loss impact plasmonic enhancement of the emission? 3. Plasmons and nonlinear optics –a winning combination? •Why and how do the metals absorb and reflect? •Can a metal be made lossless? •Does it have to be a metal? •Can metal loss be compensated by gain? •Can one make a true sub-wavelength laser /spaser? RBNI-14 15 Surface charge oscillations +- +- -+ +- +- +- -+ +- -+ +- +- +m<0 -+ -+ +- +- +- +- -+ -+ +- +- +- +-+ -+ -+ +- +- +- +- -+ -+ +- +- +- Surface plasmon (SP) d>0 m (sp ) d (sp ) 0 p sp 1 D RBNI-14 16 Interface surface plasmon polaritons (SPP) Surface plasmons couple with electromagnetic waves Unfortunately…in a real metal 3 <0 spp dm c d m 2.8 SPP is a TM wave propagating along the interface and evanescent in both mediums, with dispersion Energy (eV) >0 weff c vg n 2.9 2.7 2.6 2.5 2.4 2.3 2.2 0 5 10 15 20 25 Wave number (relative units) Near the resonance interface SPP is characterized by a very small (sub-wavelength) effective width weff, large effective index c/ and small group velocity The losses are important!!!!! RBNI-14 17 Localized SPP of a sub-wavelength nanoparticle For sub-wavelength dimensions one can use electro-static approximation and solve Laplace equation in stead of wave equation r l a a r a l Pl (cos ) Emax,l l 1 ra l 1 a r M 0 + D 0 r ra a a l cos Emax,l 2 2 a ra r 0 2 Surface charge density oscillations coupled with electric field 2l 1 l 0 Emax,l Pl (cos ) l 1 Most Important is the Dipole mode l=1 p m (1 ) 2 D 0 1 2 D 1 RBNI-14 18 Dipole SPP of a sub-wavelength nanoparticle Electric Field Emax Dipole Moment Radiative Decay: p1 2 a3 0 Emax,1 3 2 2 a rad 3 D D Larger particle-larger antenna + The field is confined in a small effective volume ! Veff ,l For dipole mode Veff ,1 2 a3 D 8 a3 (l 1)2 D ~ a3 RBNI-14 Larger particle – less field concentration 19 Scope •Why are the metals necessary for sub-wavelength confinement? •What are the surface plasmons and polaritons? •Why does subwavelength confinement in optical range always means high loss? •Why reducing loss is so important? 3 Case studies 1. Who needs negative index? 2. How does loss impact plasmonic enhancement of the emission? 3. Plasmons and nonlinear optics –a winning combination? •Why and how do the metals absorb and reflect? •Can a metal be made lossless? •Does it have to be a metal? •Can metal loss be compensated by gain? •Can one make a true sub-wavelength laser /spaser? RBNI-14 20 The heavy price of having free carriers t=/2 In the sub-wavelength metallic structures (in all three dimensions !) half of the time almost all the energy is stored in kinetic motion of electrons –where it is being lost with the decay rate 2 of the order of 10 fs-1. Therefore the rate of energy loss in truly sub-wavelength structure ,eff, is always of the order of . J Case of propagating SPP Here sub-wavelength simply means very large wave-vector 3 2.9 vg Energy (eV) 2.8 c n 2.7 High Loss in “true plasmon” region 2.6 2.5 2.4 2.3 Long Range SPP 2.2 0 5 10 15 20 Wave number (relative units) 25 RBNI-14 21 Kinetic and Magnetic Inductances Kinetic inductance is caused by the inertia of electrons a v2 2 a I 2 I2 U K Nm A2 a LK 2 0 p2 A 2 2 J H 2 a LK ; 2 A p 0 2r E Split Ring Resonator Magnetic Inductance A-current area with skin effect LM 0 a ln U M LM The ratio of energies U K LK 8a r goes to zero as the physical size is decreased As particle gets progressively smaller more and more energy gets stored in the form of kinetic energy of electrons RBNI-14 22 Finding Effective Q of plasmonic mode Qeff a LM LK eff LK J 3 10 H 2a=/4 a=4r 2r Split Ring Resonator -example Q-factor E /8 2 Qm0 /16 10 /32 1 10 Border wavelength ~ 0 10 -1 (where optics – displacement current and electronics 10 p - conductivity current meet H j r 0 E + E electronics Au optics 0 10 1 10 b 2 10 3 10 Wavelength (m) U M 0 2a 2 ~ 2 2a 2 UE 0 r When conductivity current dominates there is no wavelength dependence between E RBNI-14 to store the energy 23 and H fields, hence H field can be large enough Estimate of effective loss Q of Drude metal structure actually gets reduced with wavelength and only starts getting better when < - THz region which is not really optics –and even then LC circuit is not a very high Q resonator!!!!! 2 10 1 10 optics eff()/ 0 10 -1 /32 10 /16 /8 -2 10 -3 2a=/4 electronics 10 -4 10 (b) 0 10 1 10 b Wavelength (m) 2 10 That is why high Q resonators in electronics include: low loss inductances with high (energy stored in spin magnetization and not kinetic energy of electrons) Quartz crystals Surface acoustic waves /2 cavities …. The true sub-wavelength region where kinetic inductance dominates (plasmonics) is inherently lossy and various geometric tricks will not mitigate this loss significantly! RBNI-14 24 Plasmonic and loss are inseparable Confinement (1/a) a /n Conductivity current Free carriers ELECTRONICS p a /n a /n Reactive current, free carriers PLASMONICS a /n a /n Reactive (Displacement) Conductivity current current, bound Bound carriers carriers Absorber OPTICS 1/100nm 1/50m RBNI-14 Frequency (n/) Reactive current of free carriers can be confined on sub wavelength scale but then it cannot engender magnetic field strong enough to store the energy without excessive loss 25 Scope •Why are the metals necessary for sub-wavelength confinement? •What are the surface plasmons and polaritons? •Why subwavelength confinement in optical range always means high loss? •Why reducing loss is so important? 3 Case studies 1. Who needs negative index? 2. How does loss impact plasmonic enhancement of the emission? 3. Plasmons and nonlinear optics –a winning combination? •Why and how do the metals absorb and reflect? •Can a metal be made lossless? •Does it have to be a metal? •Can metal loss be compensated by gain? •Can one make a true sub-wavelength laser /spaser? RBNI-14 26 What are plasmonic metal nanostructures good for? 1. High field concentration near the metal-dielectric interface enhance linear and nonlinear optical properties , such as electroluminescence, photoluminescence, Raman effect and so on. 2. High effective index and strong confinement of interface SPP can be used to develop sub-wavelength passive and active photonic devices (nano-laser and “spaser”) 3. New “meta-materials” with unusual properties including negative refractive index 0 0 n 0 =377 0 Bending with no reflection – perfect lensing RBNI-14 27 Case study 1: Who needs negative index? Why do we need negative index material? Presumably to get to the super-resolution……-Pendry’s superlens. n -n n Perfect near field focus object But we know that the moment we introduce finite loss in the metal , the resolving power of this near field lens deteriorates drastically image 150 1 p2 100 j 2 50 Z(nm) 1 2 p 2 0 -50 -100 -150 RBNI-14 0 X(nm)) 10 20 28 30 What if we reduce (or eliminate) loss? Create “artificial dielectric” with metal nanoparticles playing the role of atoms with large polarizability 02 Q eff d f 2 f d 0 2 j Q(1 2 / 02 ) j 02 p2 / K Resonant frequency (K=3 for spheres) f-filling fraction Q / Near the resonance we can get fairly large values of 1/2 neff ~ eff ~ Q1/2 Then we can just make a conventional lens with high resolution /neff that would give us a magnified image in the far field RBNI-14 29 What does it mean? n -n n 4 object image metal - dielectric composite lens 3 Silver 2 superlens 1 6 3 1 -1 Loss(s ) With the current high metal loss the Superlens does not really perform 13 0.2 x 10 If the loss is reduced one can get higher resolution and magnified image in far field without negative n! Reduced loss would change everything! RBNI-14 30 Meta-Catch 22 A catch-22 is a paradoxical situation in which an individual cannot avoid a problem because of contradictory constraints or rules. Often these situations are such that solving one part of a problem only creates another problem, which ultimately leads back to the original problem. As long as metal loss stays what it is, negative index materials in optical range are highly unfeasible If one could reduce metal loss by an order of magnitude or more the negative index materials and devices may become highly unnecessary Probably makes sense to see what we can do about loss RBNI-14 31 Case study 2: how loss impacts plasmonic enhancement of emission, absorption, Raman…. RBNI-14 32 Plasmonic Enhancement of Radiative Efficiency-why loss is critical Surface-plasmon-enhanced light emitters based on InGaN quantum wells KOICHI OKAMOTO, ISAMU NIKI1, ALEXANDER SHVARTSER, YUKIO NARUKAWA,TAKASHI MUKAI2 AND AXEL SCHERER Nature Materials 3, 601 (2004) >10 –fold enhancement (some form the enhanced absorption) RBNI-14 33 Origin of plasmonic enhancement Radiative decay is proportional to the density of states 1 rad Oscillator strength e f 3D 5 103 f 3D m3 / ps 2 3 D 2 0m0 2 n 4 2 3 3 c 3 2 About 1/THz.m3 @400nm 1 nrad ~ 3D phonon ~ 4 Radiative time even for the allowed transition~100ps 3 ph ph Nonradiative time can be much shorter…because density of final states is orders of magnitude higher rad RBNI-14 1 rad nrad 1 1 nrad rad rad nrad 34 “The bottleneck” Density of states Nonradiative processes rad radiation 1 rad nrad 1 1 rad nrad nrad rad RBNI-14 35 Removing bottleneck Nonradiative states 1 nrad 1 rad 3D photon states 3D 4 Purcell Factor: rad 1 FP ~ ~ r1 3 D 3 1 r11 rad 1 ~ 1 V High density “photon” states How to increase photon density? 1. Reduce the volume V<<λ3 or 2. Reduce the frequencies range Q=/>>1 Original 1946 Paper – 28 lines, 5 equations RBNI-14 36 “Dense photons” Density of states can be changed only by coupling to other media (polariton effect) or by coupling between counter-propagating photons (microcavity, photonic crystals) Propagating “dense photons” – polaritons, slow light, SPP FP ~ gap P ~ cvg1 3 D 0 vg=d/dk c k Photon states Polariton states 0 resonance can be atomic, Bragg… Dense States are Slow States – Do not couple well into true photons RBNI-14 37 “Dense photons II” Density of states can be changed only by coupling to other media (polariton effect) or by coupling between counter-propagating photons (microcavity, photonic crystals) Localized “dense photons” – microcavity, localized SP Veff n-cavity resonance cav 3 FP ~ ~ 3D Veff Free photon states Dense States are Confined Do not couple well into true photons RBNI-14 38 Re-emergence of the bottleneck Nonradiative states 1 nrad 1 rad 1 r11 FP rad 1 ~ 1 V 3D photon states 3D rad 4 3 Transfer from “dense” (high impedance) to normal (low impedance) photons “dense photons” nrad Nonradiative decay of “dense photons” More nonradiative states RBNI-14 39 The bottleneck is alive and well…just shifted down the line Density of states Reservoir of dense photons rad Nonradiative processes radiation nrad radiation Nonradiative processes Nonradiative decay RBNI-14 40 So…what is the enhancement? Nonradiative states 1 nrad 1 rad 3D photon states Original radiative efficiency 1 rad rad Purcell enhancement 1 r11 FP rad “dense photons” nrad Out-coupling efficiency out New radiative efficiency rad ,1 out 1 FP rad 1 1 FP rad nrad rad nrad 1 1 nrad rad rad nrad rad nrad rad F1 Overall enhancement rad ,1 out out rad rad (1 rad ) FP1 rad Good emitter cannot be improved!!!! RBNI-14 41 Trade offs Use effective index ' / k p p D as a parameter 104 3 2.9 Purcell Factor 10 2.8 Energy (eV) 2 Group Index=c/vg Loss 100 Effective width -2 10 1 2 3 4 Effective Index ’ More of a Photon 5 vg c n 2.7 2.6 2.5 2.4 2.3 2.2 0 5 10 15 20 25 Wave number (relative units) More of a Plasmon The tight confined and slow SPP offer large enhancement of radiation rate, but they are lossy and difficult to couple RBNI-14 outside 42 Main result for Propagating Interface SPP F1 Substantial improvement can be achieved only for very inefficient emitters with rad<1% RBNI-14 43 Enhancement with localized SPP Dipole Moment p1 2 a3 0 Emax,1 Larger particle-better antenna + Effective Volume a3 Veff D Smaller particle – better resonator It is easy to see that the same particle cannot be a good antenna and a good resonator at once! Only the emitters whose efficiency is originally quite poor can be significantly enhanced by SPP Similarly, when it comes to absorption only very weak absorbers can be enhanced by SPP Maximum possible enhancement is on the order of RBNI-14 Q rad nrad 44 Case study 3: Plasmons and nonlinear optics RBNI-14 45 Rationale: Nonlinear optical interactions are quite interesting and important, yet are also very weak – how can one improve it? It is well known that if one used pulsed (mode-locked) laser and concentrate the same average power into the high peak power with low duty cycle (d.c) efficiency of nonlinear processes will increase P t Can we do the same in the space domain and concentrate the same power into higher local power density to increase the efficiency ? Ag Plasmonics as a ”silver bullet” for nonlinear optics RBNI-14 “Mode-locking in space?” 46 Plasmonic concentrators - + - - + + + - + + + + + + ( ) 1 2 j + + M. Stockman, P. Nordlander - + - + + + - + - + Q~ r ~ ~ 10 20 i 2 Elocal 4 4 5 ~ Q ~ 10 10 E 2 Elocal 2 2 3 ~ Q ~ 10 10 E - p2 But: Plasmonic concentration always brings loss RBNI-14 47 Practical figure of merit Switching I pump f For nonlinear switching using XPM or SPM fnl I signal c Ln2 I ~ 2 L nmax ~ For wavelength conversion ~ c 2 Ln2 I pump ~ 2 2 nmax ~ 1 Maximum interaction length is determined by absorption hence the ultimate figure of merit is what is the a maximum phase shift achievable : fmax ~ 2 And how close it is to 1… L RBNI-14 nmax 48 Enhancing nonlinear index I pump Ag I sig c(3) E pump Ag Ag Ag Ag Esig f – volume filling factor Ag Ag f Ag – mode overlap RBNI-14 49 Assessing nonlinearity enhancement c eff(3) n2,eff 4 3 4 ~ 12 f Q ~ 10 ~ 12 f Q n2 c (3) This sounds mighty good….. What about absorption? Maximum phase shift eff 2 nd fmax 2 eff Enhanced as much as few hundreds times still, assuming 3 fQ n2,eff I ~ 4 Q 3n2 I This sounds really good…..except n2 1013 cm2 / W (chalcogenide glass) fnl ,max 1010 I indicating that the input pump pump density must be in excess of 10GW/cm2 in order to attain switching or efficient frequency conversion, meaning that while the length of the device can get reduced manyfold, the switching power cannot and remains huge…. and the things only go further downhill from here on once it is realized that all of the enhancement is achieved because the pump field is really concentrated by a factor of of 1000 GW/cm2 –way past break down!50 RBNI-14 Q2 >100! Local “intensity” is now in excess Saturation of nonlinearity Even if one disregards optical damage nonlinear index change saturates Borchers et al, “Saturation of the all-optical Kerr effect in solids”, Opt Lett 37, 1541 (2012) So, what is the real limit? RBNI-14 51 A slightly better figure of merit Re( psig ,nl ) Psig ,nl ~ 12 0 f Q c 4 (3) 2 E pump Esig 6 f Q 0 ( nlocal Esig 2 Factor of Q2 makes perfect sense –because SPP mode is a harmonic oscillator with a given Q – changing local index shifts resonant frequency and causes change in polarizability proportional to Q2 0 0 Assuming that maximum index change is limited by material properties to nlocal nmax 0.01 the maximum phase shift is… nl ,max ~ 2 eff 3 f Q 2 nmax Qnmax 0.01 The path to achieve either all-optical switching or efficient frequency conversion is RBNI-14 52 less than obvious Phase shift vs. distance 0 10 -1 1m2 13 n2 10 cm / W P=1W 2 Phase Shift (rad) 10 f=10-3 -2 10 -3 10 -4 f=10-4 1m2 f=10-5 n2 1013 cm2 / W 10 -5 10 P=1W f=10-6 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 Length (cm) RBNI-14 53 Two ways to define figure of merit Scientific approach --- + + + + -- What is the maximum attainable enhancement of nonlinear susceptibility? + + + + For c(2) enhancement is fQ3 ~102-103 For c(3) enhancement is fQ6 ~105-106 + + -- + + + + Engineering approach For the nonlinear index type process – what is the maximum phase shift attainable at 10dB loss? What would be the overall maximum attainable result at ~one absorption length? max~Qnmax~10-2<< Not enough for all-optical switch (or frequency conversion) RBNI-14 54 Why such a conflicting result ? Scientific approach: what matters is the relative improvement Take very weak process with efficiency approaching 0….then if the end result is <<1 Result a very large power = 10 0 Engineering approach: what matters is the end result Result = 0 × 10 Ag a very large power << 1 Using metal nanoparticles for enhancement of second order nonlinear processes may not be a “silver bullet” we are looking for. Plasmonic enhancement is an excellent technique for study of nonlinear optical properties (the higher order the better) and sensing using it, but it perhaps less stellar for any type efficient switching, conversion, gating etc. RBNI-14 55 Scope •Why are the metals necessary for sub-wavelength confinement? •What are the surface plasmons and polaritons? •Why subwavelength confinement in optical range always means high loss? •Why reducing loss is so important? 3 Case studies 1. Who needs negative index? 2. How does loss impact plasmonic enhancement of the emission? 3. Plasmons and nonlinear optics –a winning combination? •Why and how do the metals absorb and reflect? •Can a metal be made lossless? •Does it have to be a metal? •Can metal loss be compensated by gain? •Can one make a true sub-wavelength laser /spaser? RBNI-14 56 Ec How do the metals absorb? E Conduction Band 2 There have to be Real transitions into the Real states Intraband transitions –assisted by scattering R 2 ( finitial f final all states 3 EF 2 1 ( eE k ) m 2 c 4 2 2 M scat (E final Einitial ) Involves the states within from the Fermi level At small frequencies only the states near Conduction Fermi level are involved , but at optical Band 1 frequencies –all bets are off and everything k depends on density of final states 3 Valence Band Im( ) ~ ( ) 2 2 ( ) ~ M scat final 3 At even higher frequencies also InterbandRBNI-14 transitions –band edge 57 Why are the metals lossier than Semiconductors? E E E Semiconductor Metal final ,met EF EF final ,sem R ~ final k Valence Band k Valence Band Density of final states is much higher in the metals Of course plasma frequency is higher in metals, hence trade–off is inevitable, but semiconductor is a viable alternative in theRBNI-14 THz regime 58 Why at optical frequencies loss does not decrease dramatically with temperature? E To maintain energy conservation phonons must be involved To maintain momentum conservation Phonons must have wave vector commensurate with the Fermi wave vector ph ph (kF ) For low frequencies only absorption of phonons is possible – this process is ph (kF ) proportional to the number of phonons and thus highly temperature ~ 1 5THz dependent ph EF kF k Valence Band RBNI-14 For high frequencies both absorption and emission of phonons is possible – the later process has a spontaneous component that does not depend on phonon number and thus is temperature independent 59 Temperature independent absorption with e-escattering (b) (a) k2 E E k2 Hee h Two electrons are excited by a single photon Hee k4 h EF EF k3 k1 k3 k1 Only Umklapp scattering is contributing k E ( k3 k1 ( k4 k2 g k (c) E (d) ee ( ) ~ ( / E F k2 h k4 k2 h k4 2 EF EF Hee k3 Hee k1 k3 k k1 k RBNI-14 The process becomes very important at high frequencies 60 How does the metal reflects-refracts? E Virtual transitions to real states Conduction Band 2 Intraband transitions-onto itself Im( ) ~ Rv ~ ( f initial f initial k ( eE ) 2 k2 Looks very Drude-like Involves only the states near the Fermi level initial states Positive detuning Ec 2 Negative detuning Interband transitions EF Conduction Band 1 k Valence Band ( ) ( ) ib ( ) j 3 2 p 2 ib ~ initial states 2 f ( Pif E / 2 ( E final Einitial 0 Involves all the states below the Fermi level p2 e2 f v2f / 3 0 Depends only on Fermi-level properties Depends only on density of states 2 2 ( ) ~ M scat final on all final states RBNI-14 61 Scope •Why are the metals necessary for sub-wavelength confinement? •What are the surface plasmons and polaritons? •Why subwavelength confinement in optical range always means high loss? •Why reducing loss is so important? 3 Case studies 1. Who needs negative index? 2. How does loss impact plasmonic enhancement of the emission? 3. Plasmons and nonlinear optics –a winning combination? •Why and how do the metals absorb and reflect? •Can a metal be made lossless? •Does it have to be a metal? •Can metal loss be compensated by gain? •Can one make a true sub-wavelength laser /spaser? RBNI-14 62 Decoupling absorption and reflection E Conduction Band 2 There is no final state for the Absorption to take place p2 ( ) ( ) ib ( ) 2 j 3 E12 Ec p2 e2 f v2f / 3 0 ( ) ~ EF Conduction Band 1 Ecv Valence Band 2 2 M scat final 0 So intraband absorption is 0, but what about interband? Still 0 because there are no real states in the gap RBNI-14 63 Lossless, yet metal? E Conduction Band 2 There is no final state for the absorption to take place Ec Ecv , E12 E12 Ec But we still want negative p2 ( ) ib ( ) 2 EF Conduction Band 1 Ecv Thus we want narrow bands with wide gaps Valence Band 2 p2 e2 f v2f / 3 0 We need large Fermi velocity (small effective mass) As always in Nature we see two contradictory demands and thus should see if some type of trade-off and compromise is possible RBNI-14 64 Consider b.c.c. lattice (Na) Tight Binding Model a Vss Coupling strength Vss Ec EF Brillouin Zone –f.c.c. RBNI-14 65 Lossless metal condition 1/2 16Vss 0.6e Vss / a 0 ib 2 E12 p Solution is possible but it take place at large interatomic distances that are bigger than in typical metals (and also prone to Mott localization) Ec Lossy dielectric Dielectric Lossless metal Lossy metal 0 Vss~exp(-a/a0) a0/a a RBNI-14 66 Example:Na 0 -1 -2 3p Energy (eV) -3 -4 Ef -5 E12 Ec 3s -6 -7 -8 -9 -10 2 4 6 8 10 12 14 Lattice Constant (A) Need Lattice constant of 8 Angstrom instead of 4.3 Angstrom That is why metals absorb…. RBNI-14 67 Hypothetical 8 A lattice spacing Na 3 Interband Absorption 2 1 0.6 Free Carrier Absorption 0.4 i 0.2 0 -1 -2 -3 12 1500 p c 2000 2500 r 3000 Wavelength (nm) RBNI-14 68 Big question: what do we do? • We need to use stoichiometric arrangement of metal atoms separated by the non-metal atoms • ITO –like materials are not solution – need stoichiometry! • Example: AlO –metal • Maybe 2D structures? O Al Al Zn AlZnO2 RBNI-14 69 69 Prospective materials Na7MgF8 Na F Mg Na F Mg Mg Na 9.7eV Mg F Mg RBNI-14 70 70 Scope •Why are the metals necessary for sub-wavelength confinement? •What are the surface plasmons and polaritons? •Why subwavelength confinement in optical range always means high loss? •Why reducing loss is so important? 3 Case studies 1. Who needs negative index? 2. How does loss impact plasmonic enhancement of the emission? 3. Plasmons and nonlinear optics –a winning combination? •Why and how do the metals absorb and reflect? •Can a metal be made lossless? •Does it have to be a metal? •Can metal loss be compensated by gain? •Can one make a true sub-wavelength laser /spaser? RBNI-14 71 Potential and Kinetic energy of the oscillator Equation of motion Amplitude e / m 0 r0 ( ) 2 0 2 j d 2r dr e 2 r E 0 dt 2 dt m Dielectric constant Ne2 / m 0 () 1 2 0 2 j Potential Energy of electrons UP Nm02 r0 4 2 Kinetic Energy of electrons 1 Ne2 / m 0 2 2 0 E 0 2 2 2 4 ( 0 2 2 UK Electric field Energy Nm 2 r0 4 Ne / m ( 1 ( 1 0 2 2 UV U P U K UE 0 E 0 E 1 2 2 2 2 2 4 4 (0 Far from resonance 0 ( ) 1 2 0 2 1 UV 0 E 2 4 2 1 2 Ne / m 0 UP 0E 4 02 Ne2 / m 0 1 Ne2 / m 0 2 U E 0 E 1 UV U P 2 4 0 UP 1 Ne2 / m 0 2 2 0E 2 4 (02 2 2 2 “Pure” Electric field Energy 2 UV 2 02 UK 0 All of the electric energy is potential ( UM RBNI-14 0 0 72 In the dispersive region 2 2 2 2 ( 1 1 Ne / m Ne / m 2 2 0 0 0 Electric field Energy U E 0 E 0 E 1 2 2 2 2 2 2 2 2 2 2 4 4 (0 (1 0 Potential Energy of electrons 2 “Non-dispersive (static)contribution” 4 ( 0 E 1 Ne / m 0 1 U P 0 E 202 0 E 2 2 4 (02 2 2 2 4 2 Ne / m 0 Ne / m 0 2 2 2 2 2 2 2 2 ( 2 2 0 (0 2 2 UP,S Kinetic Energy of electrons UP,D 1 Ne2 / m 0 2 2 UK 0E 2 2 2 4 ( 0 2 2 “Equal dispersive (dynamic)contributions” With dispersion – additional energy storage in dipole oscillations –less need for magnetic field 0 2 0 UV UP,S RBNI-14 UP,D UM UK 73 In the Reststrahlen region ( Balance (0 UV UP,S UP,D UM UK 0 0 Since 1 U P ,S UV E 2 0 4 U P,D U K U M 0? If we add the second dielectric with >0 UP UV UP,S UP,D UP,2 UM UK <0 2>0 No different from metal: UP In metal 0=0 hence there is no potential energy RBNI-14 UM UK 74 Improvement? What if we use dielectric, such as SiC in Reststrahlen region (10-12 microns) 2 2 LO TO 1 2 2 j TO Re( ) TO QM 1 QM 2 2 Im( ) <0 LO TO SiC >0 <0 SiO2 >0 Interface phonon polariton decay constant is a few ps!!!! (not 10fs as in metal) The width of SPP resonance eff~ gets reduced relative to the metal as it depends on QM1 It means that phonon polariton decays in time slower than plasmon polariton But the propagation length depends on QM2< QM1 hence the propagation length is not much longer(slow light effect) 1 Lprop ( d / d eff Similarly, field enhancement is proportional to QM2< QM1 hence it is not as high as in metals RBNI-14 75 U ~ ( / E 2 Purcell effect is weak Another way? <0 Introduce highly dispersive dielectric component: Rb atoms, Quantum dots into the metaldielectric plasmonic structure….. Ag Ag >0 SiO2 <0 SiO2 ( >0 0 0 Now significant amount of energy is in the form o fkinetic energy of bound electrons – less energy goes into conduction electrons oscillations-less scattering UM UM- Magnetic energy UV- “pure E-field energy” UP,S- “non-dispersive potential energy of bound electrons UP,D- “highly-dispersive potential UP,S UP,D energy of bound electrons UV RBNI-14 UK,M-Kinetic Energy of electrons in metal UK,D UM UK,M UK,D-Kinetic Energy of bound electrons 76 “Slow-light loaded plasmon poariton” Most Energy is contained in the oscillations of bound electrons in dielectric – less in the oscillations of conduction electrons in the metal The width of SPP resonance eff gets reduced relative to the metal It means that SL loaded SPP decays in time slower than SPP with non-dispersive dielectric But the propagation length does not change at all in SL loaded SPP (slow light effect) Lprop ( d / d eff1 Similarly, field enhancement is SL-loaded SPP is no better than in normal SPP U ~ ( / E 2 RBNI-14 77 Can metal loss be compensated by gain? RBNI-14 78 Scope •Why are the metals necessary for sub-wavelength confinement? •What are the surface plasmons and polaritons? •Why subwavelength confinement in optical range always means high loss? •Why reducing loss is so important? 3 Case studies 1. Who needs negative index? 2. How does loss impact plasmonic enhancement of the emission? 3. Plasmons and nonlinear optics –a winning combination? •Why and how do the metals absorb and reflect? •Can a metal be made lossless? •Does it have to be a metal? •Can metal loss be compensated by gain? •Can one make a true sub-wavelength laser /spaser? RBNI-14 79 Two relevant questions (1)Can one compensate the SP loss with gain? <0 loss >0 The rate of loss in the metal is on the scale of 1/(10 fs) – it can only be compensated by the gain medium which has high density of active atoms and allowed transitions semiconductor gain Semiconductor Can one make a small “Nanolaser” or Spaser? “spaser” Metal loss Semiconductor gain RBNI-14 80 Purcell Factor Spontaneous decay is proportional to the density of photon states 8 1 ~ = In 3D space spon 3D 3 Metal Veff 1 1 Localized SPP mode ~ ~ LSP LSP For localized SP Veff spon LSP 3 3 Purcell Enhancement FP,LSP ~ ~ ~ Q ~ 102 104 LSP 3D Veff Veff <0 >0 Metal deff vg = ∂ω ∂β Dielectric For propagating SPP Propagating SPP mode Purcell Enhancement FP,LSP d eff spon LSP 3 ~ ~ ~ 101 103 SPP 3D deff vg But is it good or bad? RBNI-14 1 SPP ~ SPP ~ 81 Can one compensate the plasmon loss with gain? Rate equations <0 Ag Density of electron loss >0 gain Semiconductor current Purcell’s Factor dnel nel I Fp dt eVeff rad g a(nel ntr ) gain differential gain ntr ~ 1019 cm 3 Veff 3 / 20 Fp ~ 100; rad ~ 10 10 s 1 J tr ~ eVeff1/3ntr FP rad ~ MA / cm 2 Transparency density It is the current density and not the carrier density that matters. Purcell Enhancement does play the role of a spoiler – not surprising because in lasers we always want to reduce spontaneous emission and not enhance it RBNI-14 82 Compensation of loss in propagating SPP E z M<0 2 z E S 2/qS >0 Ez - Loss J Gain g J Weff x n-doped N Wa kx Rsp Egap p-doped x=2/kx + Weff-90% energy effective width normalized to /n neff kx /n n / c x Effective modal index Weff Lx RBNI-14 83 How does it look? 107 106 Transparency current goes up as(1+neff5) (11/neff2) Transparency current density 105 10 1 0 1019 eff (s-1) 10180 1014 neff 1013 0 2.2 1.8 0.1 “Real Plasmon” Ntr(cm-3) Fp 1040 100 0.2 0.3 0.4 0.5 0.6 Purcell’s factor 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Long Range SP Jtr(A/cm2) Metal: Silver. To fit the SPP to the bandgap for wide range of wavelengths we need to use “hypothetical” InxGaxNyAs1-y semiconductor 0.7 Weff Transparency carrier density 0.1 0.2 0.3 0.4 0.5 0.6 Modal loss 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.7 0.1 500 Weff Purcell’s factor goes up as (1+neff5) Transparency density is reasonable As expected, loss goes up as (1-1/neff2) Weff Effective modal index 1.4 10 Weff 0.2 0.3 0.4 RBNI-14 0.5 0.6 0.7 Weff (nm) 84 Scope •Why are the metals necessary for sub-wavelength confinement? •What are the surface plasmons and polaritons? •Why subwavelength confinement in optical range always means high loss? •Why reducing loss is so important? 3 Case studies 1. Who needs negative index? 2. How does loss impact plasmonic enhancement of the emission? 3. Plasmons and nonlinear optics –a winning combination? •Why and how do the metals absorb and reflect? •Can a metal be made lossless? •Does it have to be a metal? •Can metal loss be compensated by gain? •Can one make a true sub-wavelength laser /spaser? RBNI-14 85 Enter The Spaser LASER Light Amplification by Stimulated Emission of Radiation SPASER Surface Plasmon Amplification by Stimulated Emission of Radiation Stockman, 2003 RBNI-14 86 N-doped AlGaAs Is Spaser Unique? ++ + + + + d Intrinsic GaAs Barrier AlAs ( d gd Ag barrier 10nm Very small amount of radiation is coming out for small SPASER Energy is mostly contained in the matter, not field Barrier AlAs Fraction of energy contained in free electron oscillations P-doped AlGaAs na 2 d f M 1 d gd gd 1 fD d gd na 2 1 Total fraction of energy in the electron vibrations: d gd 1 fe d gd Fraction of energy contained in bound electron oscillations in dielectric (polariton) For 40 nm GaAs Ag Spaser operating at 550nm RBNI-14 fe 0.8 87 Is Spaser Unique? + + + + R>99% + Fraction of energy contained in free electron oscillations Consider a small semiconductor laser with high reflectivity mirrors Very small amount of radiation is coming out of it Fraction of energy contained in 1 bound electron oscillations in f gs S s gs dielectric (polariton) gs 1 Total fraction of energy in the electron vibrations: f fM 0 e s gs For GaAs laser operating at 880 nm –strong dispersion! ( s (ns gs s 2ns s 2nng ~ 22 f 0.60 Energy is mostly contained e in the matter, not field In any laser operating in a dispersive material not a photon but a polariton is emitted and most of the energy is contained in electronic vibrations! The only difference is that in the RBNI-14 88 SPASER it is free electrons. So what’ s so special? Big loss! Spaser and VCSEL N-doped AlGaAs Intrinsic GaAs Barrier AlAs Ag Barrier AlAs barrier 10nm P-doped AlGaAs VCSEL MATTER SPASER Bound Electrons Energy in semiconductor 36% 60% Free Electrons Energy in metal 30% 10% 45% Magnetic Field Energy Electric Field Energy FIELD RBNI-14 3% 16% 89 A very simple way to look at the spaser…. Holes flow Spontaneous radiation Rate equation for SP – bosonic particle dN SP g eff ( N SP 1) loss N SP dt The threshold is defined as Metal loss N SP 1 Equal probability of stimulated and spontaneous emission (linewidth is decreased by a factor of 2) geff loss / 2 g Semiconductor gain loss metal rad Spontaneous SPP emission –energy transfer from semiconductor to metal is at the rate of loss Electrons flow Current is I ~ e loss ~ 20 A This best case result does not depend on shape, size( as long as it is truly sub-wavelength, i.e. single mode) or on the gain material Current density 6 J ~ I / πr 2 ~ 10RBNI-14 A / cm 2 90 What happens? We want to reduce the current by reducing the volume 1 Itr ~ eVeff ntr FP rad Unfortunately…Purcell factor is inversely proportional to the volume The current scales down with the volume only until the dimensions become sub-wavelength – after that – the same current has to go into progressively ever smaller volume –not a very good idea! The current is proportional to the number of modes (confined and free space) into which the gain medium emits. The less is the volume the less is the number of modes. Until you have just one mode left. Then the volume stops being important. RBNI-14 91 80 nm Single mode Au-InGaAs Spaser “emitting” at 1320 nm Active layer P-contact Au N-contact 15 10 14 b 10 loss c γeff -effective linewidth d 13 10 c 8 b 6 4 a 2 12 10 -7 10 -6 10 -5 10 It -4 10 Current (A) Input output characteristics shows no threshold 0 0 It 100 200 Current (A) Carrier density (gain) is clamped near threshold –sign of “spasing” Linewidth narrowing indicates “spasing threshold” of 28 A RBNI-14 300 8 6 4 threshold a Number of SPP’s in the mode loss 12 10 d “output”RSP Carrier density (1018cm-3) SPP emission rate (1/s) and Linewidth (1/s) 10 2 0 0 2 4 Carrier density 6 8 Nc,th (1018cm-3) Near threshold half of SP’s are coherent and half have random phases – so it more like “SPED” rather than “SPASER” 92 10 Lineshape evolution 1 -1 -2 -3 -4 -5 (a) -6 0.7 0.8 0.9 Energy (eV) 0.5 Material Gain (1014 s-1) 0 -0.5 -1 -1.5 -2 -2.5 (b) -3 0.7 1 0.8 1 Below threshold Low pump 1.5 0.5 0 -0.5 -1 -1.5 (c) -2.5 0.7 0.8 0.9 Energy (eV) At threshold 1 Material Gain (1014 s-1) Modal gain spectral density (a.u) 1.5 1 -2 0.9 Energy (eV) Modal gain spectral density (a.u) Material Gain (1014 s-1) 0 Modal gain spectral density (a.u) Modal gain spectral density (a.u) 1 0.5 0 -0.5 -1 -1.5 (d) -2 0.7 1 RBNI-14 0.8 0.9 Energy (eV) 1 Way above threshold 93 The threshold current depends on Scattering Rate in Metal The threshold current does not depend on •Shape of the Mode •Size of the mode as long as it is sub-wavelength in all 3 Dimensions •Confinement factor of the mode •Gain material! (as long as it works in a “normal laser”) •Temperature (weakly) •Wavelength (red, orange, IR, polka-dot) as long it is less than ~20m •Longitude, Latitude, Altitude •Attitude of the scientist •Amount of money spent Ithr = eγeff nsp ~ 20μA RBNI-14 94 Do we need a nanolaser? • True sub-diffraction structure always require metal and thus inherently very lossy • Therefore, threshold of the sub-diffraction laser will always be very high (20 A in a tiny volume) • There will be only a few photons (plasmons) in a mode – no coherence (linewidth of THz) • But the device will be very fast (THz) What if we just use spontaneous emission – SPED (Surface Plasmon Emitting Diode)? RBNI-14 95 Characteristics of SPED • • • • Small volume –high density of integration Low power consumption Easier to cool High speed due to Purcell Effect RBNI-14 96 Linewidth (Hz) Intracavity Power (W) VCSEL, SPASER and SPED 10-2 VCSEL 10-3 (a) SPASER -4 10 LED SPED 10-5 -6 10 10-7 10-8 102 103 104 105 106 Pump Current density (A/cm2) 1014 1012 (b) 1010 10 1013 1012 108 107 108 107 108 SPASER LED 8 VCSEL 106102 Maximum frequency (Hz ) SPED 107 103 104 105 106 Pump Current density (A/cm2) (c) SPASER VCSEL 11 10 1010 9 10 108102 LED 103 SPED 104 105 106 RBNI-14 Pump Current density (A/cm2) 97 Conclusions: •Loss affects and limits everything that is purportedly good about plasmonics and metamaterials Reducing loss would change the paradigm. •To get sub-wavelength concentration of light in all three dimensions one does need material with <0 (usually a metal) •In optical and near IR region it always leads to losses commensurate with rate of decay in metal -1/10 fs, but far IR and THz the situation is less dire –perhaps mid-IR I swhere the action should be •Primarily because of loss only very weak optical processes can be enhanced by plasmonic means (SERS) – any optical device that requires high efficiency cannot be improved by plasmonic means. Hence sensing is the most promising niche. •Loss in metals in optical range is fundamentally different from loss (resistance) at low frequencies. •Loss and negative dielectric constant can be decoupled (in the future) •Using phonon polaritons and other resonant schemes reduces loss but has drawbacks of its own •Compensating loss with gain is tricky and probably unrealistic •Incoherent sub-wavelength sources are just as good as lsub-waveelngth lasers and might have a future. RBNI-14 98 Additional reading J. B. Khurgin, G. Sun “In search of elusive lossless metal, Appl. Phys. Lett, 96, 181102 (2010) J. B. Khurgin, G. Sun, Enhancement of optical properties of nanoscaled objects by metal nanoparticles”, J. Opt. Soc. B, 26, 83 (2009) J. B. Khurgin, G. Sun, Impact of high-order surface plasmon modes of metal nanoparticles on enhancement of optical emission, Appl. Phys. Lett., 95, 171103 (2009) J. B. Khurgin, G. Sun, “Enhancement of light absorption in quantum well by surface plasmon polariton”, Appl. Phys. Lett, 94, 191106 (2009) J. B. Khurgin, G. Sun, R. A. Soref, “Plasmonic enhancement of photoluminescence by metal nanoparticles”, Appl. Phys. Lett. 94, 101103 (2009); J. B. Khurgin, G. Sun, R. A. Soref, “Practical limits of absorption enhancement near metal nanoparticles”, Appl. Phys. Lett. 94, 071103 (2009) G. Sun, J.B. Khurgin, R. A. Soref, “Plasmonic light-emission enhancement with isolated metal nanoparticles and their coupled arrays “, J. Opt. Soc. Am. B 25, 1748 (2008) J.B. Khurgin, G. Sun, R. A. Soref “Electroluminescence efficiency enhancement using metal nanoparticles”, Appl. Phys. Lett., 93 021120 (2008) Khurgin JB, Sun G, Soref RA ”Enhancement of luminescence efficiency using surface plasmon polaritons: figures of merit” J. Opt. Soc. Am. B 24: 1968-1980 (2007) Khurgin JB ”Surface plasmon-assisted laser cooling of solids”, Phys. Rev. Lett 98, Art. No. 177401 (2007) Sun G, Khurgin JB, Soref RA ”Practicable enhancement of spontaneous emission using surface plasmons” Appl. Phys. Lett., RBNI-14 90 Art. No. 111107 (2007) 99 J. B. Khurgin, A. Boltasseva, “Reflecting upon the losses in plasmonics and metamaterials”, MRS Bulletin , 37, 768-779 (2012) J. B. Khurgin , G. Sun, “Practicality of compensating the loss in the plasmonic waveguides using semiconductor gain medium”, Appl. Phys. Lett, 100, 011105 (2012) J.B. Khurgin and G. Sun: “How small can “ Nano ” be in a “ Nanolaser ”?”, Nanophotonics, 1, 3-8 (2012) J. B. Khurgin, G. Sun, “Injection pumped single mode surface plasmon generators: threshold, linewidth, and coherence”, Optics Express 20 15309-15325 (2012) G. Sun, J. B. Khurgin, “Origin of giant difference between fluorescence, resonance, and nonresonance Raman scattering enhancement by surface plasmons”, Phys. Rev. A 85, 063410 (2012 JG. Sun, J. B. Khurgin, and D. P. Tsai, “Comparative analysis of photoluminescence and Raman enhancement by metal nanoparticles”, Opt. Letters, 37, 1583-1585 (2012) G. Sun, J. B. Khurgin, A. Bratkovsky “Coupled-mode theory of field enhancement in complex metal nanostructures “, Phys. Rev, B 84, 045415 (2011) J. B. Khurgin, G. Sun, Scaling of losses with size and wavelength in Nanoplasmonics” Appl. Phys. Lett, 99, 211106 (2011) B. Zhang, J. B. Khurgin, “Eigen mode approach to the sub-wavelength imaging with surface plasmon polaritons” Appl. Phys. Lett, 98, 263102 (2011) G. Sun, J. B. Khurgin, Optimization of the nanolens consisting of coupled metal nanoparticles: An analytical approach”, Appl. Phys. Lett, 98, 153115 (2011) G. Sun, J. B. Khurgin, “Plasmon Enhancement of Luminescence by Metal Nanoparticles : IEEE J. of Selected Topics In Quantum Electronics , 17 ,110 (2011) J. B. Khurgin, G. Sun , “Theory of optical emission enhancement by coupled metal nanoparticles: An analytical approach”, Appl. Phys. Lett 98, 113116 (2011) G. Sun, J. B. Khurgin, Comparative study of field enhancement between isolated and coupled metal nanoparticles: An analytical approach”, Appl. Phys. Lett. 97, 263110 (2010) RBNI-14 100 Extra RBNI-14 101 Higher order modes High order modes tend to cling to the surface they are non-radiative l=2 l=5 l=10 Veff ,l 8 a3 (l 1)2 D Small volume– high field concentration E r l 1 r a a E1 ~ e j1t Pl (cos ) Emax,1 l 2 a ra r r RBNI-14 a 102 Impedance mismatch In electronics in order to go from the high impedance circuit element (output stage of power amplifier-10K ) to the low impedance medium (coaxial cable -75, strip line) one uses Impedance transformer e.g. source follower VDD vin IREF vout -VSS Now we want “dense photons” to serve as a buffer between the emitting medium and the propagating photons Can this buffer be lossless? RBNI-14 103 Impedance mismatch In electronics in order to go from the high impedance circuit element (output stage of power amplifier-10K ) to the low impedance medium (coaxial cable -75, strip line) one uses Impedance transformer e.g. source follower VDD vin IREF vout -VSS Now we want “dense photons” to serve as a buffer between the emitting medium and the propagating photons Can this buffer be lossless? RBNI-14 104 Threshold of a laser with just one mode Bjork, G., Karlsson, A. & Yamamoto, Y. Definition of a laser threshold. Phys. Rev. A 50, 16751680 (1994). We propose that the threshold of a laser is more appropriately described by the pump power (or current) needed to bring the mean cavity photon number to unity, rather than the conventional“ definition" that it is the pump power at which the optical gain equals the cavity loss. In general the two definitions agree to within a factor of 2, but in a class of microcavity lasers with high spontaneous emission coupling efficiency and high absorption loss, the de6nitions may differ by several orders of magnitude. J. B. Khurgin, G. Sun, “Injection pumped single mode surface plasmon generators: threshold, linewidth, and coherence”, Optics Express 20 15309-15325 (2012) ..it corresponds to the number of SPs at threshold roughly Nsp,t~1 i.e., on average, just about one SP in the mode, which is the threshold definition according to [36].” RBNI-14 105 Single mode spaser as a “threshold-less” laser Rate equations Effective linewidth eff loss geff Excess noise factor ~1 dN SP ( g eff loss N SP g eff nspon dt dN c e 1 I ex g eff N SP g eff nsp rspon rA Carriers dt SP Auger recombination (proportional to volume) Excitation current Spontaneous emission into other modes If we neglect emission in other modes and nonradiative recombination… Rate of SP generation is RSP NSP / loss I ex / e N SP I ex / e loss No threshold! Since due to small volume SP emission into the single mode is the dominant recombination mechanism, all the energy goes into SP’s, but are they coherent? We choose the critical effective linewidth narrowing It corresponds to geff ,c loss / 2 and Critical (threshold?) current eff ,c loss geff ,c loss / 2 N SP,c nsp 1 one SP in the mode I cr e eff nsp ~ 20 A RBNI-14 106